Method of estimating oxygen storage capacity of catalyst

ABSTRACT

An engine system for a vehicle includes an internal combustion engine having an exhaust gas outlet, an exhaust system having a three-way catalyst and a switch-type post oxygen sensor, and an engine control module that controls the engine system. The engine control module includes a first control logic for estimating a three-way catalyst oxygen storage capacity based on a plurality of measured inputs, a second control logic for estimating aging effects of the switch-type post oxygen sensor, and a third control logic that calculates a filtered estimated three-way catalyst oxygen storage capacity for the three-way catalyst.

INTRODUCTION

The present disclosure relates generally to a method of estimating theoxygen storage capacity of a catalyst of a catalytic converter for aninternal combustion engine of a vehicle.

The ability to accurately estimate the oxygen storage capacity of athree-way catalyst results in increased fuel savings for an internalcombustion engine. Current methods of estimation of oxygen storagecapacity utilizing fuel cut off during a deceleration maneuver does notprovide an accurate enough estimation to allow for more aggressive fuelstrategy that provides such fuel savings. As a result, a new method ofestimating oxygen storage capacity is required to achieve significantfuel saving without adding hardware to the engine system.

In addition, the catalyst must work properly and at a certain capacityto effectively reduce emissions and to pass vehicle regulations.Monitoring of the catalyst's ability to function accomplishes thisobjective.

Accordingly, there is a need for a new method of estimating oxygenstorage capacity for effective fuel strategy for increased fuelefficiency and monitoring of its ability to function without addingadditional cost in vehicle hardware.

SUMMARY

In an exemplary aspect, an engine system for a vehicle includes aninternal combustion engine having an exhaust gas outlet, an exhaustsystem having a three-way catalyst and a switch-type post oxygen sensor,and an engine control module having a control logic sequence thatincludes a first control logic for estimating a three-way catalystoxygen storage capacity based on a plurality of measured inputs using:

$\frac{d\delta}{dt} = {k^{f}\left( {{\left( {\lbrack{CO}\rbrack + \left\lbrack H_{2} \right\rbrack - {2\left\lbrack O_{2} \right\rbrack}} \right)\left( {1 - {{abs}(\delta)}} \right)} - {k^{b}\delta}} \right)}$

where [CO], [H2], and [O2] are CO, H2, and O2 concentrations at thethree-way catalyst outlet and K^(f) and K^(b) are calibration constants;a second control logic for estimating aging effects of the switch-typepost oxygen sensor, and a third control logic that calculates a filteredestimated three-way catalyst oxygen storage capacity for the three-waycatalyst.

In another exemplary aspect, the control logic sequence furthercomprises a fourth control logic configured to control the internalcombustion engine based upon the filtered estimated three-way catalystoxygen storage capacity.

In another exemplary aspect, the second control logic estimates agingeffects of the switch-type post oxygen sensor using:

${\tau_{\lambda}\frac{{d\delta}_{\tau}}{dt}} = {\delta - {\delta_{\tau}.}}$

Where τ_(A) is switch-type post oxygen sensor dynamic response time

In another exemplary aspect, the first control logic estimates thethree-way catalyst oxygen storage capacity by normalizing using:(−1≤δ_(τ)≤1).

In another exemplary aspect, the control logic sequence further includesa control logic that determines the switch-type post oxygen sensordynamic response time by integrating a rich-to-lean and a lean-to-richresponse of the switch-type post oxygen sensor.

In another exemplary aspect, the first control logic further determinesan estimated switch-type post oxygen sensor voltage using:V _(λ) =f(δ_(τ));(0≤V _(λ) ≤V _(λ) _(max) ).

In another exemplary aspect, the plurality of measured inputs include atleast one of a pre-catalyst equivalence ratio, a fuel flow rate, exhaustgas pressure, a pre-catalyst exhaust gas temperature, oxygen sensorvoltage, a metered mass air flow value, an engine speed value, acatalyst temperature and a fuel control state value.

The above features and advantages and other features and advantages ofthe present disclosure are readily apparent from the following detaileddescription when taken in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWING

The drawings described herein are for illustration purposes only and arenot intended to limit the scope of the present disclosure in any way.

FIG. 1 is a schematic of an exemplary engine system in accordance withthe present disclosure;

FIG. 2 illustrates a one-dimensional portion of a three-way catalyst inthe system of FIG. 1;

FIG. 3 is a schematic representation of an exemplary three-way catalystobserver model in accordance with the present disclosure;

FIG. 4 is an exemplary flowchart illustrating a method in accordancewith the present disclosure;

FIG. 5 is a graph illustrating an exemplary performance of a three-waycatalyst observer in an engine system in accordance with the presentdisclosure; and

FIG. 6 is a graph illustrating an exemplary response of a switch-typepost oxygen sensor.

DESCRIPTION

The following description is merely exemplary in nature and is notintended to limit the present disclosure, application, or uses. The term“about” as used in the description is defined as an amount around aspecific number that does not have a significant impact on the resultsof the operation.

Referring to FIGS. 1 and 2, a schematic for an engine system 10 for avehicle is illustrated and will now be described. The engine system 10includes an internal combustion engine (ICE) 12, an exhaust system 14,and an engine control module 15. The exhaust system 14 includes acatalyst assembly 16 and an oxygen sensor 18. More particularly, thecatalyst assembly 16 has an exhaust gas inlet 20 and an exhaust gasoutlet 22, and a three-way catalyst 24. The oxygen sensor 18 is disposedin the exhaust gas outlet 22 and may be a switch-type post oxygensensor. The exhaust gas inlet 20 receives exhaust gas from the ICE anddirects the exhaust gas to the three-way catalyst 24. The three-waycatalyst 24 includes a ceramic substrate 26 on which is disposed acatalytic metal coating 28. In the present example, the catalytic metalcoating 28 includes Cerium Oxide (Ce₂O₃). However, other metal oxides orcombinations of metal oxides may be incorporated into the three-waycatalyst 24 without departing from the scope of the present disclosure.For example, the catalytic metal coating 28 may include oxides ofRhodium (Rh), Palladium (Pd), and Platinum (Pt) among other metaloxides.

The engine control module 15 is preferably an electronic control devicehaving a preprogrammed digital computer or processor, control logic,memory used to store data, and at least one I/O peripheral. The controllogic includes a plurality of logic routines for monitoring,manipulating, and generating data. The engine control module 15 controlsthe plurality of actuators, pumps, valves, and other devices associatedwith the engine system 10 control according to the principles of thepresent disclosure. The control logic may be implemented in hardware,software, or a combination of hardware and software. For example,control logic may be in the form of program code that is stored on theelectronic memory storage and executable by the processor. The enginecontrol module 15 receives the output signal of each of several sensorson the vehicle, performs the control logic and sends command signals toseveral control devices. For example, a control logic implemented insoftware program code that is executable by the processor of the enginecontrol module 15 includes a control logic for implementing a methoddescribed further below.

The present disclosure provides an improvement upon a three-way catalystoxygen storage capacity real-time observer that is described inco-pending, co-assigned U.S. patent application Ser. No. 16/560,361 thedisclosure of which is hereby incorporated by reference in its entirety.The three-way catalyst oxygen storage models described in U.S. patentapplication Ser. No. 16/560,361 may also be used together with theimplementation of the present disclosure.

For the purposes of the present disclosure, the three-way catalyst isvirtually separated into a plurality of segments 30. One such segment31, is shown in FIG. 2 and represents a one-dimensional portion throughwhich the catalytic reactions occur. The constituents of the exhaust gasgoing into the segment includes [O₂]_(in), [CO]_(in), [CO₂]_(in),[H₂]_(in), and [H₂O]_(in) at an incoming gas temperature T_(gin). Afterthe catalytic reaction, the treated gas coming out of the segmentincludes [O₂]_(out), [CO]_(out), [CO₂]_(out), [H₂]_(out), and[H₂O]_(out) at an outgoing gas temperature T_(gout). For example, afirst catalytic reaction is an Oxygen storage reaction represented bythe following:

O₂ + 2Ce₂O₃ ↔ 2Ce₂O₄;r₁ = k₁^(f)OSC²(1 − φ_(O₂))²[O₂] − k₁^(b)OSC²φ_(O₂)²C₀;${k_{1}^{f} = {A_{1}^{f}e^{- \frac{E_{1}^{f}}{T}}}},\mspace{14mu}{and}$$k_{1}^{b} = {A_{1}^{b}{e^{- \frac{E_{1}^{b}}{T}}.}}$

A second catalytic reaction is a Carbon Monoxide Oxidation reactionrepresented by the following:

CO + Ce₂O₄ ↔ CO₂ + Ce₂O₃;r₂ = k₂^(f)OSCφ_(O₂)[CO] − k₂^(b)OSC(1 − φ_(O₂))[CO₂];${k_{2}^{f} = {A_{2}^{f}e^{- \frac{E_{2}^{f}}{T}}}},\mspace{14mu}{and}$$k_{2}^{b} = {A_{2}^{b}{e^{- \frac{E_{2}^{b}}{T}}.}}$

A third catalytic reaction is a Hydrogen Oxidation reaction representedby the following:

H₂ + Ce₂O₄ ↔ H₂O + Ce₂O₃;r₃ = k₃^(f)OSCφ_(O₂)[H₂] − k₃^(b)(1 − φ_(O₂))[H₂O];${k_{3}^{f} = {A_{3}^{f}e^{- \frac{E_{3}^{f}}{T}}}},{and}$$k_{3}^{b} = {A_{3}^{b}{e^{- \frac{E_{3}^{b}}{T}}.}}$

Oxygen storage value (OSV) is calculated using the following equation,where OSC is the oxygen storage capacity:

${{OSC}\frac{\partial{\varphi 0}_{2}}{\partial t}} = {{2r_{1}} - r_{2} - {r_{3}.}}$

The treated exhaust gas constituents coming out of the catalyst segmentare calculated as follows:

$\left\lbrack O_{2} \right\rbrack_{out} = {{\frac{\left\lbrack O_{2} \right\rbrack_{in} + {k_{1}^{b}{OS}C^{2}{\varphi_{O_{2}}}^{2}C_{O}t_{r}}}{1 + {k_{1}^{f}{{OSC}^{2}\left( {1 - \varphi_{O_{2}}} \right)}^{2}t_{r}}}\lbrack{CO}\rbrack}_{out} = {{\frac{\lbrack{CO}\rbrack_{in} + {\left( {\lbrack{CO}\rbrack_{in} + \left\lbrack {CO}_{2} \right\rbrack_{in}} \right)k_{2}^{b}{{OSC}\left( {1 - \varphi_{O_{2}}} \right)}t_{r}}}{1 + {k_{2}^{b}{{OSC}\left( {1 - \varphi_{O_{2}}} \right)}^{2}t_{r}} + {k_{2}^{f}{OSC}_{\varphi_{O_{2}}}t_{r}}}\left\lbrack {CO}_{2} \right\rbrack}_{out} = {{\frac{\left\lbrack {CO}_{2} \right\rbrack_{in} + {\left( {\lbrack{CO}\rbrack_{in} + \left\lbrack {CO}_{2} \right\rbrack_{in}} \right)k_{2}^{b}{{OSC}\left( {1 - \varphi_{O_{2}}} \right)}t_{r}}}{1 + {k_{2}^{b}{{OSC}\left( {1 - \varphi_{O_{2}}} \right)}t_{r}} + {k_{2}^{f}{OSC}_{\varphi_{O_{2}}}t_{r}}}\left\lbrack H_{2} \right\rbrack}_{out} = {{\frac{\left\lbrack H_{2} \right\rbrack_{in} + {\left( {\left\lbrack H_{2} \right\rbrack_{in} + \left\lbrack {H_{2}O} \right\rbrack_{in}} \right)k_{3}^{b}{{OSC}\left( {1 - \varphi_{O_{2}}} \right)}t_{r}}}{1 + {k_{3}^{b}{{OSC}\left( {1 - \varphi_{O_{2}}} \right)}t_{r}} + {k_{3}^{f}{OSC}_{\varphi_{O_{2}}}t_{r}}}\left\lbrack {H_{2}O} \right\rbrack}_{out} = \frac{\left\lbrack {H_{2}O} \right\rbrack_{in} + {\left( {\lbrack{CO}\rbrack_{in} + \left\lbrack {CO}_{2} \right\rbrack_{in}} \right)k_{3}^{f}{{OSC}\left( {1 - \varphi_{O_{2}}} \right)}t_{r}}}{1 + {k_{3}^{b}{{OSC}\left( {1 - \varphi_{O_{2}}} \right)}t_{r}} + {k_{3}^{f}{OSC}_{\varphi_{O_{2}}}t_{r}}}}}}}$

Turning now to FIG. 3, a three-way catalyst observer model 32 isillustrated and will now be described. The three-way catalyst observermodel 32 includes a Kalman filter 34 and a three-way catalyst kineticmodel 36. The three-way catalyst kinetic model 36 includes a switch-typepost lambda sensor model 38. Inputs 40 into the three-way catalystobserver model 32 include, for example, a pre-catalyst equivalenceratio, a fuel flow rate, exhaust gas pressure, a pre-catalyst exhaustgas temperature, oxygen sensor 18 voltage, a mass air flow value, anengine speed value, a catalyst temperature and a fuel control statevalue. Outputs 42 of the three-way catalyst observer model 32 include anoxygen storage value (OSV), a post-catalyst equivalence ratio (EQR), apost-catalyst switch voltage, an oxygen storage capacity (OSC), and apre-catalyst EQR offset.

Turning now to FIG. 4 with continuing reference to FIG. 3, a flowchartis illustrated for a method 50 of dynamically estimating the OSC of thethree-way catalyst 24. The method 50 includes a first step 52 ofdetermining oxygen ion responsiveness to exhaust gases species with anormalized variable to represent oxygen ion concentrations in the sensorand while estimating aging effects on the sensor using:

${\frac{d\delta}{dt} = {k^{f}\left( {{\left( {\lbrack{CO}\rbrack + \left\lbrack H_{2} \right\rbrack - {2\left\lbrack O_{2} \right\rbrack}} \right)\left( {1 - {{abs}(\delta)}} \right)} - {k^{b}\delta}} \right)}};\left( {{- 1} \leq \delta \leq 1} \right)$${{\tau_{\lambda}\frac{{d\delta}_{\tau}}{dt}} = {\delta - \delta_{\tau}}};\left( {{- 1} \leq \delta_{\tau} \leq 1} \right)$

Where τ_(λ) is switch-type post oxygen sensor dynamic response time

Where [CO], [H2], and [O2] are CO, H2, and O2 concentrations at thethree-way catalyst outlet using a three-way catalyst model (an exampleof which is described previously) and K^(f) and K^(b) are calibrationconstants.

The method 50 continues to step 54 where the switch-type lambda sensoroutput voltage is estimated using:V _(λ) =f(δ_(τ)); (0≤V _(λ) ≤V _(λ) _(max) )

The method 50 then continues to step 56 where the observer uses a Kalmanfilter to correct the estimated oxygen storage and then calculates thethree-way catalyst oxygen storage capacity.

With reference to FIG. 5, a graph 60 depicting the performance of thethree-way catalyst observer model 32 and method 50 is illustrated andwill now be described. The graph 60 includes a y-axis 62 depicting theestimated OSC over time 64 in seconds (x-axis). The dashed referencelines represent Worst Performing Acceptable (WPA) mean 66, WPA −4σ 68,Best Performing Unacceptable (BPU) +2σ 70 (70 is for BPU with a WPA postO2 sensor), and BPU mean 72. The performance lines represent calculatedtime-based WPA mean 74, time-based WPA −4σ 76, time-based BPU +2σ 78,and time-based BPU mean 80. The vertical dashed line represents theequivalent time of two Federal Test Procedure (FTP) cycles 82. Theseveral lines contained by the bracket 83 represent the WPA degradedpost oxygen sensor. The several lines contained by the bracket 84represent the non-degraded post oxygen sensor. The bracket 86 representsthe difference between the WPA −4σ and BPU +2σ.

Estimated OSV is used to determine fuel strategy. For example, whenestimated OSV is low, a lean fuel strategy (air/fuel ratio is less thanstoichiometric) is incorporated to introduce less fuel into the engine.Less fuel requires less Oxygen to burn the fuel leaving more Oxygen tostore in the catalyst. Alternatively, stoichiometric and rich air/fuelratio leaves less Oxygen available to store in the catalyst andtherefore the oxidation of CO and H2 in the catalyst depletes the Oxygenstorage of the catalyst. Current fuel strategies do not have the inputof an accurate OSV estimation and therefore are required to assume OSVis low and requires more Oxygen to increase storage leading to reducedengine performance and higher fuel consumption. The capability to have amore accurate OSV estimation allows engine calibration to moreaccurately determine when the catalyst requires Oxygen to increase OSVand therefore run a fuel strategy more tailored to engine performanceand other parameters that fuel strategy is used to control.

The oxygen storage capacity of the catalyst 24 is an indicator of theability of the catalyst to effectively reduce emissions. For example, ifthe catalyst has aged to a significant extent, the oxygen storagecapacity will be low and the catalyst can be deemed to be insufficientto perform its emission reduction function when then oxygen storagecapacity is below a threshold. In addition, if the wrong type ofcatalyst is installed in a vehicle, it may also not meet the thresholdoxygen storage capacity, which would also indicate that the catalyst isnot function property. Therefore, the present system is configured tosend a signal indicating that the oxygen storage capacity is below thethreshold, so that corrective action may be taken. For example, thesignal may be used actuate a malfunction light, such as a “check engine”light. In addition, or in the alternative, the signal may be used by thevehicle controller to perform other corrective actions, such as limitingthe vehicle's fuel supply until the catalyst is replaced and meets theoxygen storage capacity minimum threshold.

Referring now to FIG. 6, a graph 600 illustrates the response of aswitch-type post lambda sensor. The responsiveness of a switch-type postlambda sensor depends upon the age of the sensors. In general, oldersensors have a slower response. The responsiveness may be determinedfrom two calibration tables for engine fueling processes for each of arich-to-lean transition and a lean-to-rich transition. The horizontalaxis 602 of the graph 600 corresponds to time and the vertical axis 604corresponds to the voltage from the switch-type post lambda sensor. Theinputs to the table are integrated values for each of the rich-to-leantransition 606 and the lean-to-rich transition 608. This process may beperformed during an engine fuel cut off response, for example, to obtaina sensor response that accounts for the effects of aging on theswitch-type post oxygen sensor. In this manner, the actual sensorresponse which may have changed over time may be determined and may thenbe used to account for the aging effects on sensor responsiveness in theabove-described method and system. This, in turn, provides the abilityto improve the estimation of the oxygen storage capacity of thethree-way catalyst.

While examples have been described in detail, those familiar with theart to which this disclosure relates will recognize various alternativedesigns and examples for practicing the disclosed method within thescope of the appended claims.

The following is claimed:
 1. An engine system for a vehicle, the enginesystem comprising: an internal combustion engine having an exhaust gasoutlet; an exhaust system having a three-way catalyst and a switch-typepost oxygen sensor; and an engine control module having a control logicsequence, and wherein the engine control module controls the enginesystem and the control logic sequence includes: a first control logicfor estimating a three-way catalyst oxygen storage capacity based on aplurality of measured inputs using:${\frac{d\delta}{dt} = {k^{f}\left( {{\left( {\lbrack{CO}\rbrack + \left\lbrack H_{2} \right\rbrack - {2\left\lbrack O_{2} \right\rbrack}} \right)\left( {1 - {{abs}(\delta)}} \right)} - {k^{b}\delta}} \right)}};$where [CO], [H2], and [O2] are CO, H2, and O2 concentrations at thethree-way catalyst outlet and K^(f) and K^(b) are calibration constants;a second control logic for estimating aging effects of the switch-typepost oxygen sensor; and a third control logic that calculates a filteredestimated three-way catalyst oxygen storage capacity for the three-waycatalyst.
 2. The system of claim 1, wherein the control logic sequencefurther comprises a fourth control logic configured to control theinternal combustion engine based upon the filtered estimated three-waycatalyst oxygen storage capacity.
 3. The system of claim 1, wherein thesecond control logic estimates aging effects of the switch-type postoxygen sensor using:${\tau_{\lambda}\frac{{d\delta}_{\tau}}{dt}} = {\delta - {\delta_{\tau}.}}$Where τ_(λ) is switch-type post oxygen sensor dynamic response time. 4.The system of claim 1, wherein the first control logic estimates thethree-way catalyst oxygen storage capacity by normalizing using:(−1≤δ_(τ)≤1).
 5. The system of claim 1, wherein the control logicsequence further includes a control logic that determines theswitch-type post oxygen sensor dynamic response time by integrating arich-to-lean and a lean-to-rich response of the switch-type post oxygensensor.
 6. The system of claim 1, wherein the first control logicfurther determines an estimated switch-type post oxygen sensor voltageusing:V _(A) =f(δ_(τ)); (0≤V _(λ) ≤V _(λ) _(max) ).
 7. The system of claim 1,wherein the plurality of measured inputs include at least one of apre-catalyst equivalence ratio, a fuel flow rate, exhaust gas pressure,a pre-catalyst exhaust gas temperature, oxygen sensor voltage, a meteredmass air flow value, an engine speed value, a catalyst temperature and afuel control state value.
 8. An engine system for a vehicle, the enginesystem comprising: an internal combustion engine having an exhaust gasoutlet; an exhaust system having a three-way catalyst and a switch-typepost oxygen sensor, and wherein the exhaust system includes an exhaustgas inlet in downstream communication with the exhaust gas outlet of theinternal combustion engine; and an engine control module adapted to:estimate of the oxygen storage capacity of the three-way catalyst basedon a plurality of measured inputs using:$\frac{d\delta}{dt} = {k^{f}\left( {{\left( {\lbrack{CO}\rbrack + \left\lbrack H_{2} \right\rbrack - {2\left\lbrack O_{2} \right\rbrack}} \right)\left( {1 - {{abs}(\delta)}} \right)} - {k^{b}\delta}} \right)}$where [CO], [H2], and [O2] are CO, H2, and O2 concentrations at thethree-way catalyst outlet and K^(f) and K^(b) are calibration constants;estimate a voltage output for the switch-type post oxygen sensor; andcorrect the estimated oxygen storage capacity based upon a comparisonbetween the estimated voltage output for the switch-type post oxygensensor and an actual voltage output for the switch-type post oxygensensor.
 9. The system of claim 8, wherein the engine control module isfurther adapted to control the internal combustion engine based upon thecorrected three-way catalyst oxygen storage capacity.
 10. The system ofclaim 8, wherein the engine control module is further adapted estimateaging effects of the switch-type post oxygen sensor using:${{\tau_{\lambda}\frac{{d\delta}_{\tau}}{dt}} = {\delta - \delta_{\tau}}},$Where τ_(λ) is switch-type post oxygen sensor dynamic response time. 11.The system of claim 8, wherein the engine control module estimates theoxygen storage of the three-way catalyst by normalizing using:(−1≤δ_(τ)≤1).
 12. The system of claim 8, wherein the engine controlmodule further determines a switch-type post oxygen sensor dynamicresponse time by integrating a rich-to-lean and a lean-to-rich responseof the switch-type post oxygen sensor.
 13. The system of claim 8,wherein engine control module estimates the voltage output for theswitch-type post oxygen sensor using:V _(λ) =f(δ_(τ)); (0≤V _(λ) ≤V _(λ) _(max) ).
 14. The system of claim 8,wherein the plurality of measured inputs include at least one of apre-catalyst equivalence ratio, a fuel flow rate, exhaust gas pressure,a pre-catalyst exhaust gas temperature, oxygen sensor voltage, a meteredmass air flow value, an engine speed value, a catalyst temperature and afuel control state value.
 15. A method of estimating an oxygen storagecapacity of a three-way catalyst in an engine system for a vehicleincluding an internal combustion engine having an exhaust gas outlet,and an exhaust system having a three-way catalyst and a switch-type postoxygen sensor, the method comprising: estimating a three-way catalystoxygen storage capacity based on a plurality of measured inputs using:${\frac{d\delta}{dt} = {k^{f}\left( {{\left( {\lbrack{CO}\rbrack + \left\lbrack H_{2} \right\rbrack - {2\left\lbrack O_{2} \right\rbrack}} \right)\left( {1 - {{abs}(\delta)}} \right)} - {k^{b}\delta}} \right)}};$where [CO], [H2], and [O2] are CO, H2, and O2 concentrations at thethree-way catalyst outlet and K^(f) and K^(b) are calibration constants;estimating aging effects of the switch-type post oxygen sensor; andcalculating a filtered estimated three-way catalyst oxygen storagecapacity for the three-way catalyst.
 16. The method of claim 15, whereinestimating the three-way catalyst oxygen storage capacity furthercomprises normalizing using:(−1≤δ_(τ)≤1).
 17. The method of claim 15 further comprising controllingthe internal combustion engine based upon the filtered estimatedthree-way catalyst oxygen storage capacity.
 18. The method of claim 15further comprising estimating aging effects of the switch-type postoxygen sensor using:${{\tau_{\lambda}\frac{{d\delta}_{\tau}}{dt}} = {\delta - \delta_{\tau}}},$Where τ_(λ) is switch-type post oxygen sensor dynamic response time. 19.The method of claim 15, further comprising determining the switch-typepost oxygen sensor dynamic response time by integrating a rich-to-leanand a lean-to-rich response of the switch-type post oxygen sensor. 20.The method of claim 15, further comprising determining an estimatedswitch-type post oxygen sensor voltage using:V _(λ) =f(δ_(τ)); (0≤V _(λ) ≤V _(λ) _(max) ).